Seminar| Institute of Mathematical Sciences
Time：Tuesday, Feburary 14th, 2023, 13:30-14:30
Location：RS408, IMS
Speaker: Pengyu Yang, Chinese Academy of Sciences
Abstract: Dirichlet’s approximation theorem describes how precise one can approximate a real vector by rational vectors. In 1969 Davenport and Schmidt defined Dirichlet improvable vectors, for which Dirichlet’s approximation theorem can be improved. They showed that Lebesgue almost every vector in R^n is non-improvable, and asked if the same conclusion holds if one restricts to a curve in R^n. We will show that if the affine span of the curve satisfies certain Diophantine/arithmetic conditions, then almost every point on the curve is non-improvable. Our approach is based on Dani correspondence, which relates the problem to equidistribution of evolution of curves in homogeneous spaces. Joint work with Nimish Shah.